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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.16.0002.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, c z] HypergeometricPFQRegularized[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], r]}, {Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], s]}, d z] == Sum[(Product[Pochhammer[Subscript[a, j], m] c^m, {j, 1, p}]/ Product[Gamma[Subscript[b, j] + m] m!, {j, 1, q}]) (Product[Pochhammer[Subscript[\[Alpha], j], k - m] d^(k - m), {j, 1, r}]/ Product[Gamma[Subscript[\[Beta], j] + k - m] (k - m)!, {j, 1, s}]) z^k, {k, 0, Infinity}, {m, 0, k}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", RowBox[List["c", " ", "z"]]]], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Alpha]", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Beta]", "s"]]], "}"]], ",", RowBox[List["d", " ", "z"]]]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "m"]], "]"]], " ", SuperscriptBox["c", "m"]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], " ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "+", "m"]], "]"]], " ", RowBox[List["m", "!"]]]]]]], FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Alpha]", "j"], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", SuperscriptBox["d", RowBox[List["k", "-", "m"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "s"], " ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Beta]", "j"], "+", "k", "-", "m"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]]]]], SuperscriptBox["z", "k"]]]]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a p ; b 1 , , b q ; c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", RowBox[List["c", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] r F ~ s ( α 1 , , α r ; β 1 , , β s ; d z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["r", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["s", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", SubscriptBox["\[Beta]", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", RowBox[List["d", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 m = 0 k ( j = 1 p ( a j ) m TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "m"], Pochhammer] ) ( j = 1 r ( α j ) k - m TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Alpha]", "j"], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] ) d k - m c m z k ( j = 1 q Γ ( b j + m ) ) ( j = 1 s Γ ( β j + k - m ) ) m ! ( k - m ) ! p F ~ q ( a 1 , , a p ; b 1 , , b q ; c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", RowBox[List["c", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] r F ~ s ( α 1 , , α r ; β 1 , , β s ; d z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["r", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["s", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", SubscriptBox["\[Beta]", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", RowBox[List["d", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 m = 0 k ( j = 1 p ( a j ) m TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "m"], Pochhammer] ) ( j = 1 r ( α j ) k - m TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Alpha]", "j"], ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] ) d k - m c m z k ( j = 1 q Γ ( b j + m ) ) ( j = 1 s Γ ( β j + k - m ) ) m ! ( k - m ) ! [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", RowBox[List["c_", " ", "z_"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Alpha]_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Alpha]_", "r_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["\[Beta]_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Beta]_", "s_"]]], "}"]], ",", RowBox[List["d_", " ", "z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "m"]], "]"]], " ", SuperscriptBox["c", "m"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Alpha]", "j"], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", SuperscriptBox["d", RowBox[List["k", "-", "m"]]]]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "+", "m"]], "]"]], " ", RowBox[List["m", "!"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "s"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Beta]", "j"], "+", "k", "-", "m"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]]]]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29